Introduction

Understanding how to find the equation of a straight line that passes through specific points is a fundamental concept in algebra and geometry. This article focuses on the process of determining the value of c, the y-intercept, for a line given in the form y mx c, and passing through the points (3, 4) and (7, 10).

Deriving the Equation of the Line

Given the line equation y mx c, where m is the slope, and we need to find c. The line passes through the points (3, 4) and (7, 10).

Step 1: Calculate the slope m. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

m (y2 - y1) / (x2 - x1)

Substituting the points (3, 4) and (7, 10) into the formula:

m (10 - 4) / (7 - 3) 6 / 4 3 / 2

Step 2: The equation of the line using the slope m is:

y - 4 (3/2) (x - 3)

Simplifying this, we get:

2y - 8 3x - 9

2y 3x - 1

y (3/2)x - 1/2

Thus, we find c -1/2. This value represents the y-intercept of the line, the point where the line crosses the y-axis.

Verification

To verify, we substitute the given points back into the equation:

For (3, 4): y (3/2) 3 - 1/2 9/2 - 1/2 8/2 4

For (7, 10): y (3/2) 7 - 1/2 21/2 - 1/2 20/2 10

The line's equation satisfies both points, confirming our solution.

Conclusion

By applying basic algebraic principles, we found the line equation y (3/2)x - 1/2 that passes through the points (3, 4) and (7, 10). The value of c, the y-intercept, is -1/2. This example demonstrates the importance of understanding linear equations in both mathematical and real-world applications.

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For more articles on algebra, geometry, and calculus, visit our resource page. Understanding these fundamental concepts will enhance your mathematical toolkit and enable you to tackle more complex problems.